Why is $\forall x \in A:P(x)$ equivalent to $\forall x (x\in A \to P(x)) $?

Question

In the book that I'm studying from it defines $\forall x \in A: P(x)$ equivalent to $\forall x (x\in A \to P(x))$ without any explanation as to why it is that way. The same thing for the existential quantifier: $\exists x \in A: P(x)$ is equivalent to $\exists x(x\in A \land P(x))$.

I attempted to understand these definitions by applying to real world examples.

Example for $\forall x \in A: P(x)$ equivalent to $\forall x (x\in A \to P(x)) $:

[Example One] All cars have wheels

Let $C$ be a set of all cars and $P(x)$ stand for "$x$ has wheels."

Then the English statement analyzed in logical form is $\forall x \in C: P(x)$ is equivalent to $\forall x (x\in C \to P(x))$. Translating the statement means "If $x$ is a car, then $x$ has wheels." However, sometimes when I analyze it I get "For every $x$, $x$ is a car and $x$ has wheels." (In logical form, $\forall x(x \in C \land P(x))$) which book says it's wrong.

Again, I have the same trouble with the existential quantifier definition as well.

[Example Two] Some cars have three wheels.

Let $C$ be a set of all cars and $P(x)$ stand for "$x$ has three wheels."

Then the English statement analyzed in logical form is $\exists x \in C: P(x)$ which is equivalent to $\exists x(x \in C \land P(x))$. This translates to "There exists at least one $x$ such that $x$ is a car and $x$ has three wheels." But sometimes I translate these kinds of statements as $\exists x(x \in C \to P(x))$, which is wrong according to the book's definition.

I'm asking how can I make it clear these two definitions are different. Also, why $\forall x \in A: P(x)$ is not equivalent to $\forall x (x\in A \land P(x))$ and that $\forall x \in A: P(x)$ is equivalent to $\forall x (x\in A \to P(x))$. Similarly, why $\exists x \in A: P(x)$ is equivalent to $\exists x(x\in A \land P(x))$ and that $\exists x \in A: P(x)$ is not equivalent to $\exists x(x\in A \to P(x))$? In short, can you help make it clear to me why these definitions are defined the way they are.

Answer 1

It seems that you are confused about the role of $A$ (or $C$) in all this.

In your example 1, both expressions mean "The cars in set C all have wheels." They do not mean that all cars have wheels, and in fact, there are plenty of cars (not in C) that don't have wheels.

In your example 2, the statement means "There is a car in set C that has three wheels." However your alternate version "$\exists x, (x\in C\to P(x))$ means something different. If there is a car not in C, that car is a witness to the expression $(x\in C \to P(x))$, regardless of how many wheels that car has. This is a common source of confusion. "If p then q" is true when p is false, regardless of whether q is true or not.

Answer 2

Let $C$ be the set of all cars, and let $P(x)$ stand for "$x$ has three wheels." The statements

(1) $\exists x \in C\,P(x)$

(2) $\exists x\,(x \in C \wedge P(x))$

are equivalent by definition because (2) is defined as a shorthand for (1). The direct translations of these two statements respectively into natural language are

(1') There is a car $x$ such that $x$ has three wheels

(2') There is an (object) $x$ such that $x$ is a car and $x$ has three wheels.

In English the statements (1') and (2') are equivalent to one another and also to the simpler statement "some car has three wheels."

To see why all these existential statements are different in meaning from the corresponding universal statements, let's continue using $P(x)$ to stand for "$x$ has three wheels" and consider the statements

(3) $\forall x \in C\,P(x)$

(4) $\forall x\,(x \in C \to P(x))$.

Again (3) and (4) are equivalent by definition because (4) is defined as a shorthand for (3). The direct translations of these two statements respectively into natural language are

(3') For every car $x$, $x$ has three wheels

(4') For every (object) $x$, if $x$ is a car then $x$ has three wheels.

In English the statements (3') and (4') are equivalent to one another and also to the simpler statement "every car has three wheels."

To see why existential statements (1), (2), (1'), and (2') differ in meaning from the universal statements (3), (4), (3'), and (4'), observe that the former are true in the real world but the latter are not. In other words, some, but not all, cars have three wheels.